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Theorem pwssplit2 17146
Description: Splitting for structure powers, part 2: restriction is a group homomorphism. (Contributed by Stefan O'Rear, 24-Jan-2015.)
Hypotheses
Ref Expression
pwssplit1.y  |-  Y  =  ( W  ^s  U )
pwssplit1.z  |-  Z  =  ( W  ^s  V )
pwssplit1.b  |-  B  =  ( Base `  Y
)
pwssplit1.c  |-  C  =  ( Base `  Z
)
pwssplit1.f  |-  F  =  ( x  e.  B  |->  ( x  |`  V ) )
Assertion
Ref Expression
pwssplit2  |-  ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  ->  F  e.  ( Y  GrpHom  Z ) )
Distinct variable groups:    x, Y    x, W    x, U    x, Z    x, V    x, B    x, C    x, X
Allowed substitution hint:    F( x)

Proof of Theorem pwssplit2
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pwssplit1.b . 2  |-  B  =  ( Base `  Y
)
2 pwssplit1.c . 2  |-  C  =  ( Base `  Z
)
3 eqid 2443 . 2  |-  ( +g  `  Y )  =  ( +g  `  Y )
4 eqid 2443 . 2  |-  ( +g  `  Z )  =  ( +g  `  Z )
5 simp1 988 . . 3  |-  ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  ->  W  e.  Grp )
6 simp2 989 . . 3  |-  ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  ->  U  e.  X )
7 pwssplit1.y . . . 4  |-  Y  =  ( W  ^s  U )
87pwsgrp 15671 . . 3  |-  ( ( W  e.  Grp  /\  U  e.  X )  ->  Y  e.  Grp )
95, 6, 8syl2anc 661 . 2  |-  ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  ->  Y  e.  Grp )
10 simp3 990 . . . 4  |-  ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  ->  V  C_  U )
116, 10ssexd 4444 . . 3  |-  ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  ->  V  e.  _V )
12 pwssplit1.z . . . 4  |-  Z  =  ( W  ^s  V )
1312pwsgrp 15671 . . 3  |-  ( ( W  e.  Grp  /\  V  e.  _V )  ->  Z  e.  Grp )
145, 11, 13syl2anc 661 . 2  |-  ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  ->  Z  e.  Grp )
15 pwssplit1.f . . 3  |-  F  =  ( x  e.  B  |->  ( x  |`  V ) )
167, 12, 1, 2, 15pwssplit0 17144 . 2  |-  ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  ->  F : B --> C )
17 offres 6577 . . . . 5  |-  ( ( a  e.  B  /\  b  e.  B )  ->  ( ( a  oF ( +g  `  W
) b )  |`  V )  =  ( ( a  |`  V )  oF ( +g  `  W ) ( b  |`  V ) ) )
1817adantl 466 . . . 4  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  (
( a  oF ( +g  `  W
) b )  |`  V )  =  ( ( a  |`  V )  oF ( +g  `  W ) ( b  |`  V ) ) )
195adantr 465 . . . . . 6  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  W  e.  Grp )
20 simpl2 992 . . . . . 6  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  U  e.  X )
21 simprl 755 . . . . . 6  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  a  e.  B )
22 simprr 756 . . . . . 6  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  b  e.  B )
23 eqid 2443 . . . . . 6  |-  ( +g  `  W )  =  ( +g  `  W )
247, 1, 19, 20, 21, 22, 23, 3pwsplusgval 14433 . . . . 5  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  (
a ( +g  `  Y
) b )  =  ( a  oF ( +g  `  W
) b ) )
2524reseq1d 5114 . . . 4  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  (
( a ( +g  `  Y ) b )  |`  V )  =  ( ( a  oF ( +g  `  W
) b )  |`  V ) )
2615fvtresfn 5780 . . . . . 6  |-  ( a  e.  B  ->  ( F `  a )  =  ( a  |`  V ) )
2715fvtresfn 5780 . . . . . 6  |-  ( b  e.  B  ->  ( F `  b )  =  ( b  |`  V ) )
2826, 27oveqan12d 6115 . . . . 5  |-  ( ( a  e.  B  /\  b  e.  B )  ->  ( ( F `  a )  oF ( +g  `  W
) ( F `  b ) )  =  ( ( a  |`  V )  oF ( +g  `  W
) ( b  |`  V ) ) )
2928adantl 466 . . . 4  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  (
( F `  a
)  oF ( +g  `  W ) ( F `  b
) )  =  ( ( a  |`  V )  oF ( +g  `  W ) ( b  |`  V ) ) )
3018, 25, 293eqtr4d 2485 . . 3  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  (
( a ( +g  `  Y ) b )  |`  V )  =  ( ( F `  a
)  oF ( +g  `  W ) ( F `  b
) ) )
311, 3grpcl 15556 . . . . . 6  |-  ( ( Y  e.  Grp  /\  a  e.  B  /\  b  e.  B )  ->  ( a ( +g  `  Y ) b )  e.  B )
32313expb 1188 . . . . 5  |-  ( ( Y  e.  Grp  /\  ( a  e.  B  /\  b  e.  B
) )  ->  (
a ( +g  `  Y
) b )  e.  B )
339, 32sylan 471 . . . 4  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  (
a ( +g  `  Y
) b )  e.  B )
3415fvtresfn 5780 . . . 4  |-  ( ( a ( +g  `  Y
) b )  e.  B  ->  ( F `  ( a ( +g  `  Y ) b ) )  =  ( ( a ( +g  `  Y
) b )  |`  V ) )
3533, 34syl 16 . . 3  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  ( F `  ( a
( +g  `  Y ) b ) )  =  ( ( a ( +g  `  Y ) b )  |`  V ) )
3611adantr 465 . . . 4  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  V  e.  _V )
3716ffvelrnda 5848 . . . . 5  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  B )  ->  ( F `  a
)  e.  C )
3837adantrr 716 . . . 4  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  ( F `  a )  e.  C )
3916ffvelrnda 5848 . . . . 5  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  b  e.  B )  ->  ( F `  b
)  e.  C )
4039adantrl 715 . . . 4  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  ( F `  b )  e.  C )
4112, 2, 19, 36, 38, 40, 23, 4pwsplusgval 14433 . . 3  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  (
( F `  a
) ( +g  `  Z
) ( F `  b ) )  =  ( ( F `  a )  oF ( +g  `  W
) ( F `  b ) ) )
4230, 35, 413eqtr4d 2485 . 2  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  ( F `  ( a
( +g  `  Y ) b ) )  =  ( ( F `  a ) ( +g  `  Z ) ( F `
 b ) ) )
431, 2, 3, 4, 9, 14, 16, 42isghmd 15761 1  |-  ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  ->  F  e.  ( Y  GrpHom  Z ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   _Vcvv 2977    C_ wss 3333    e. cmpt 4355    |` cres 4847   ` cfv 5423  (class class class)co 6096    oFcof 6323   Basecbs 14179   +g cplusg 14243    ^s cpws 14390   Grpcgrp 15415    GrpHom cghm 15749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-of 6325  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-ixp 7269  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-sup 7696  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-10 10393  df-n0 10585  df-z 10652  df-dec 10761  df-uz 10867  df-fz 11443  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-plusg 14256  df-mulr 14257  df-sca 14259  df-vsca 14260  df-ip 14261  df-tset 14262  df-ple 14263  df-ds 14265  df-hom 14267  df-cco 14268  df-0g 14385  df-prds 14391  df-pws 14393  df-mnd 15420  df-grp 15550  df-minusg 15551  df-ghm 15750
This theorem is referenced by:  pwssplit3  17147
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